1. 1 problem background

With the continuous development of world sports, how to stand out from the fierce competition and win the championship in one fell swoop requires athletes not only to have super physical fitness, but also to have flexible competition skills.

In the shot put competition, how to put the shot as far as possible is closely related to physical factors such as shooting speed, shooting height, shooting angle and arm extension. Therefore, how to throw scientifically to achieve the ideal effect and satisfactory effect requires our rational analysis.

Information about 1.2 shot put

In the shot put competition, athletes are required to throw the shot with a weight of 7.257kg in a circle with a diameter of 2. 135m, as shown in the following figure:

Comprehensive analysis of the movement process of shot put can be divided into two situations:

1. Without considering the arm extension of shot put, the first mathematical model is established with the speed, height and angle of shot put as parameters.

2. Considering the throwing arm of shot put, a second mathematical model with shot put speed, shot put height, shot put angle and throwing arm as parameters is established.

3. In the whole shot put process, although air resistance always exists, its influence is very small, so it is ignored.

1.3 problems to be solved

Question 1: Taking the speed, angle and height of shot put as parameters, the mathematical model of shot put is established.

Question 2: Improve the above model by considering the athletes' action of opening their arms when putting the shot.

Question 3: On this basis, given the height of the hand, the best hand is determined for different hand speeds.

Question 4: Compare the sensitivity of throwing performance to throwing speed and angle.

5. Establishment and solution of the model

5. 1 model 1 establishment

1. Establish a throwing model with speed, angle and height as parameters;

Time of shot put from a to b:

……………… ( 1)

Maximum height of shot put:

………………(2)

All the time when the shot put falls from the height of h;

…………………(3)

Horizontal distance of shot put:

5.2 Establishment of Model 2

1. When considering the situation of athletes' arm extension, we established the second model:

In the process of arm extension, the shot put is subjected to thrust and gravity, and the force of the shot put is analyzed.

According to Newton's second law:

………………………( 1)

According to the above formula:

………………………………(2)

According to the kinematic formula:

………………………………………(3)

It can be obtained from the above formula:

……………………(4)

The above formula further shows that the hand speed is related to the angle of the hand and decreases with the increase of the angle. It is unreasonable to assume that hand speed and hand angle are independent of each other in the first model.

Similarly, we can get the distance after the shot put:

5.3 solution of model 1

1. When the height is fixed, we consider the best shooting angle at different shooting speeds, and we use Matlab7.0 to solve this problem. The solution process of is as follows:

Set =0 to calculate the value.

Because, so, then. So the best shooting angle is

At the same time, when h=0, the best shooting angle is.

5.4 Sensitivity analysis

The first and second models are mathematical models of shot put. Athletes are most concerned about how to effectively improve the shot put performance, that is, how to grasp the main factors from the three independent variables of shot put height, shot angle and shot speed to improve the shot put performance. Since the height of the shot has not changed much, it is necessary to find out the variables that have a great influence on the shot put performance from the angle and speed of the shot, that is, to compare the sensitivity of the speed and angle of the shot put.

We use Matlab7.0 software to calculate (have already calculated) sum respectively, and we can get the result.

The derivation process of is as follows:

The size of sum can be compared by Matlab7.0, and the comparison result is >; Therefore, it can be concluded that the impact of throwing speed is more obvious.